Industrial Automation (Automação de Processos Industriais)

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1 Industrial Automation (Automação de Processos Industriais) Discrete Event Systems Slides 2010/2011 Prof. Paulo Jorge Oliveira Rev Prof. José Gaspar Page 1

2 Syllabus: Chap. 5 CAD/CAM and CNC [1 week]... [2 weeks] Discrete event systems modeling. Automata. Petri Nets: state, dynamics, and modeling. Extended and strict models. Subclasses of Petri nets. Chap. 7 Analysis of Discrete Event Systems [2 weeks] Page 2

3 Some pointers to Discrete Event Systems History: Tutorial: Analyzers, and Simulators: (in Portuguese) Bibliography: * Discrete Event Systems - Modeling and Performance Analysis, Christos G. Cassandras, Aksen Associates, * Petri Net Theory and the Modeling of Systems, James L. Petersen, Prentice-Hall,1981. * Petri Nets and GRAFCET: Tools for Modeling Discrete Event Systems R. David, H. Alla, Prentice-Hall, 1992 Page 3

4 Generic characterization of systems resorting to input / output relations State equations: x ( t) y( t) in continuous time (or in discrete time) Examples? f ( x( t), u( t), t) g( x( t), u( t), t) Page 4

5 Control: Open loop vs closed-loop ( the use of feedback) Advantages of feedback? (to revisit during DES supervision study) Page 5

6 Example of closed-loop with feedback Page 6

7 Discrete Event Systems: Examples Set of events: E={N, S, E, W} Page 7

8 Discrete Event Systems: Examples Queueing systems Queue Server Clients arrival Clients departure Set of events: E= {arrival, departure} Page 8

9 Discrete Event Systems: Examples Computational Systems CPU CPU Processes Arrival CPU Processes Departure Page 9

10 Characteristics of systems with continuous variables 1. State space is continuous 2. The state transition mechanism is time-driven Characteristics of systems with discrete events 1. State space is discrete 2. The state transition mechanism is event-driven Intrinsic characteristic of discrete events systems: Polling is avoided! Page 10

11 Taxonomy of Systems Page 11

12 Levels of abstraction in the study of Discrete Event Systems Language of a chocolate selling machine : (i) Waiting for a coin. (ii) Received 1 euro coin. Chocolate A given. Go to (i). (iii) Received 2 euro coin. Chocolate B given. Go to (i). 4 sensors: Received 1 euro coin, Received 2 euro coin, Chocolate A given, Chocolate B given. Languages Timed languages Stochastic timed languages Page 12

13 Systems Theory Objectives Modeling and Analysis Design and synthesis Control / Supervision Performance assessment and robustness Optimization Applications of Discrete Event Systems Queueing systems Operating systems and computers Telecommunications networks Distributed databases Automation Page 13

14 Discrete Event Systems Typical modeling methodologies Automata GRAFCET / SFC Petri nets Augmenting in modeling capacity & complexity Page 14

15 Automata Theory and Languages Genesis of computation theory Definition: A language L, defined over the alphabet E is a set of strings of finite length with events from E. Examples: E= {α β γ} L 1 = {ε αα αβ γβα}, where ε is the null/empty string L 2 = {all strings of length 3} How to build a machine that talks a given language? or What language talks a system? Page 15

16 Operations / Properties of languages E * = Kleene-closure of E: set of all strings of finite length of E, including the null element ε. Concatenation of L a and L b : L a L b * : s E : s s s, s L, a b a a s b L b Prefix-closure of L E * : L * : s E : t st * E L Page 16

17 Operations / Properties of languages [Cassandras99] Page 17

18 Automata Theory and Languages Motivation: An automaton is a device capable of representing a language according to some rules. Definition: A deterministic automaton is a 5-tuple where: (E, X, f, x 0,F) E - finite alphabet (or possible events) X - finite set of states f - state transition function f: X x E X x 0 - initial state x 0 X F - set of final states or marked states F E [Cassandras93] Word of caution: the word state is used here to mean step (Grafcet) or place (Petri Nets) Page 18

19 Example 1 of an automaton: (E, X, f, x 0,F) E = {α β γ} X = {x, y, z} x 0 = x F = {x, z} f(x, α x f(x, β z f(x, γ z f(y, α x f(y, β y f(y, γ y f(z, α y f(z, β z f(z, γ y Page 19

20 Example 2 of a stochastic automaton (E, X, f, x 0,F) E = {α β} X = {0, 1} x 0 = 0 F = {0} f(0, α f(0, β {} f(1, α f(1, β Page 20

21 Given an automaton G = (E, X, f, x 0,F) the Generated Language is defined as L(G) := {s f(x 0,s) is defined} Note: if f is always defined then L(G)==E* and the Marked Language is defined as L m (G) := {s f(x 0,s) F} Page 21

22 Example 3: marked language of an automaton b a 0 1 b a L(G) := {ε, a, b, aa, ab, ba, bb, aaa, aab, baa, } L m (G) := {a, aa, ba, aaa, baa, bba, } Concluding, in this example L m (G) means all strings with events a and b, ended by event a. Page 22

23 Automata equivalence: The automata G 1 e G 2 are equivalent if L(G 1 ) = L(G 2 ) and L m (G 1 ) = L m (G 2 ) Page 23

24 Example 4: two equivalent automata Objective: To validate a sequence of events Page 24

25 Deadlocks (inter-blocagem) Example 5: 0 a g 1 2 b The state 5 is a deadlock. a g 3 The states 3 and 4 constitute a livelock. a b 5 4 How to find the deadlocks and the livelocks? Need methodologies for the analysis of Discrete Event Systems Page 25

26 Deadlock: in general the following relations are verified L m G L G L G m An automaton G has a deadlock if and is not blocked when L m L m G L G G L G Page 26

27 Deadlock: Example: 1 a 0 b g 2 a g 3 a b 5 4 Lm G ab, abgab, abgabgab,..., a, ab, ag, aa, aab, L G abg, aaba, abga,... L m G L G The state 5 is a deadlock. The states 3 and 4 constitute a livelock. L m G L G Page 27

28 Alternative way to detect deadlocks: Example: 0 a g 1 2 b a g 3 a b 5 4 ε g a 1 a 3 b b 2 g The state 5 is a deadlock. 4 a 0 The states 3 and 4 constitute a livelock. 3 Page 28

29 Timed Discrete Event Systems Page 29

30 Examples of Automata Classes and Applications Automaton Class Recognizable language Applications Finite state machine (FSM), e.g. Moore machines or Mealy machines Pushdown automaton (PDA) Turing machine (nondeterministic, deterministic, multitape,...) Regular languages Context-free languages Recursively enumerable languages Text processing, compilers, and hardware design Programming languages, artificial intelligence, (originally) study of the human languages Theory, complexity Page 30

31 Petri nets Developed by Carl Adam Petri in his PhD thesis in Definition: A marked Petri net is a 5-tuple (P, T, A, w, x 0 ) where: P T - set of places - set of transitions A - set of arcs A (P x T) (T x P) w - weight function w: A x 0 - initial marking x 0 :P [Cassandras93] Page 31

32 Example of a Petri net (P, T, A, w, x 0 ) P={p 1, p 2, p 3, p 4, p 5 } T={t 1, t 2, t 3, t 4 } A={(p 1, t 1 ), (t 1, p 2 ), (t 1, p 3 ), (p 2, t 2 ), (p 3, t 3 ), (t 2, p 4 ), (t 3, p 5 ), (p 4, t 4 ), (p 5, t 4 ), (t 4, p 1 )} w(p 1, t 1 )=1, w(t 1, p 2 )=1, w(t 1, p 3 )=1, w(p 2, t 2 )=1 w(p 3, t 3 )=2, w(t 2, p 4 )=1, w(t 3, p 5 )=1, w(p 4, t 4 )=3 w(p 5, t 4 )=1, w(t 4, p 1 )=1 x 0 = {1, 0, 0, 2, 0} Petri net graph t 2 p 1 t 1 p 2 p 3 p 4 p 5 3 t 4 2 t 3 Page 32

33 Petri nets Rules to follow (mandatory): Arcs (directed connections) connect places to transitions and connect transitions to places A transition can have no places directly as inputs (source), i.e. must exist arcs between transitions and places A transition can have no places directly as outputs (sink), i.e. must exist arcs between transitions and places The same happens with the input and output transitions for places Page 33

34 Alternative definition of a Petri net A marked Petri net is a 5-tuple where: P T (P, T, I, O, μ 0 ) - set of places - set of transitions I - transition input function I : T P O - transition output function O : T P μ 0 - initial marking μ 0 :P [Peterson81] Note: P = bag of places (is more general than a set of places) Page 34

35 Example of a Petri net and its graphical representation Alternative definition p 1 (P, T, I, O, μ 0 ) P={p 1, p 2, p 3, p 4, p 5 } T={t 1, t 2, t 3, t 4 } I(t 1 )={p 1 } O(t 1 )={p 2, p 3 } I(t 2 )={p 2 } O(t 2 )={p 4 } I(t 3 )={p 3, p 3 } O(t 3 )={p 5 } I(t 4 )={p 4, p 4, p 4, p 5 } O(t 4 )={p 1 } μ 0 = {1, 0, 0, 2, 0} t 2 t 1 p 2 p 3 p 4 p 5 3 t 4 2 t 3 Page 35

36 Petri nets The state of a Petri net is characterized by the marking of all places. p 1 t 1 The set of all possible markings of a Petri net corresponds to its state space. p 2 p 3 3 t 2 How does the state of a Petri net evolve? Page 36

37 Execution Rules for Petri Nets (Dynamics of Petri nets) A transition t j ε is enabled if: p i P : ( pi ) # ( pi, I ( t j )) A transition t j may fire whenever enabled, resulting in a new marking given by: p p i i # ( p i,i( t j )) # ( p i,o( t j )) #( p i, I (t j ) ) = multiplicity of the arc from p i to t j #( p i, O (t j ) ) = multiplicity of the arc from t j to p i [Peterson81 2.3] Page 37

38 Execution Rules for Petri Nets (Dynamics of Petri nets) p p #( p,i(t )) #( p,o(t )) i i i j i j [Peterson81 2.3] Page 38

39 Petri nets Example of evolution of a Petri net Initial marking: μ 0 = {1, 0, 1, 2, 0} This discrete event system can not change state.. It is in a deadlock! t 2 p 1 t 1 p 2 p 3 p 4 p t 4 2 t 3 Page 39

40 Petri nets: Conditions and Events (Places and Transitions) Example: Machine waits until an order appears and then machines the ordered part and sends it out for delivery. Conditions: a) The server is idle. b) A job arrives and waits to be processed c) The server is processing the job d) The job is complete Events 1) Job arrival 2) Server starts processing 3) Server finishes processing 4) The job is delivered Event Posconditions Preconditions - a, b c d b c d, a - 1) Job arrival 2) Start of processing 3) End of processing 4) Job is delivered b) Jobs waits processing c) Job is being processed d) Job is complete a) Server is idle Page 40

41 Discrete Event Systems Example of a simple automation system modeled using PNs t 9 An automatic soda selling machine accepts 50c and $1 coins and sells 2 types of products: SODA A, that costs $1.50 and SODA B, that costs $2.00. p 1 p t 2 1 t 2 t 3 t 4 t 5 t 7 p 4 Assume that the money return operation is omitted. p 3 t 6 p 5 t 8 p 1 : machine with $0.00; t 1,t 3,t 5,t 7 : coin of 50 c introduced; t 2,t 4,t 6 : coin of $1 introduced; t 9 : SODA A sold, t 8 : SODA B sold. Page 41

42 Petri nets: Modeling mechanisms Concurrence Conflict t 1 t 1 t 2 t 2 Page 42

43 Petri nets: Modeling mechanisms Mutual Exclusion Producer / Consumer Critical Section t 1 t 2 m Critical Section produce B consume Place m represents the permission to enter the critical section B= buffer holding produced parts Page 43

44 Petri nets: Modeling mechanisms Producer / Consumer with finite capacity s Readers / t Writers s t produce B n B' consume read n n n write Page 44

45 Extensions to Petri nets Switches [Baer 1973] f f e f e f e e Possible to be implemented with restricted Petri nets. Page 45

46 Extensions to Petri nets Inhibitor Arcs Equivalent to nets with priorities Can be implemented with restricted Petri nets? Zero tests... Infinity tests... Page 46

47 Extensions to Petri nets P-Timed nets Job arrival Start of processing End of processing Job is delivered Jobs waits processing Job is beeing processed Job is complete Server is idle Page 47

48 Extensions to Petri nets T-Timed nets Job arrival Start of processing End of processing Job is delivered Jobs waits processing Job is being processed Job is complete Server is idle Page 48

49 Extensions to Petri nets Stochastic nets Stochastic switches q 0 q 1 q 2 Transitions with stochastic timings described by a stochastic variable with known pdf μ 0 q 0 +q 1 + q 2 =1 Page 49

50 Discrete Event Systems Sub-classes of Petri nets State Machine: Petri nets where each transition has exactly one input arc and one output arc. Marked Graphs: Petri nets where each place has lesser than or equal to one input arc and one output arc. Page 50

51 Discrete Event Systems Example of DES: Manufacturing system composed by 2 machines (M 1 and M 2 ) and a robotic manipulator (R). This takes the finished parts from machine M 1 and transports them to M 2. M 1 M 2 No buffers available on the machines. If R arrives near M 1 and the machine is busy, the part is rejected. R If R arrives near M 2 and the machine is busy, the manipulator must wait. Machining time: M 1 =0.5s; M 2 =1.5s; R M1 M2=0.2s; R M2 M1=0.1s; Page 51

52 Discrete Event Systems Example of DES: Define places M 1 is characterized by places x 1 M 2 is characterized by places x 2 R is characterized by places x 3 M 1 R M 2 x 1 = {Idle, Busy, Waiting} x 2 = {Idle, Busy} x 3 = {Idle, Carrying, Returning} Example of arrival of parts: a( t) 1 0 in in {0.1, 0.7, other 1.1, 1.6, time stamps 2.5} Page 52

53 Discrete Event Systems Example of DES: Definition of events: a 1 - loads part in M 1 d 1 - ends part processing in M 1 r 1 r 2 - loads manipulator - unloads manipulator and loads M 2 M 1 R M 2 d 2 - ends part processing in M 2 r 3 - manipulator at base Page 53

54 Discrete Event Systems a 1 d t x 1 x 1 = {Idle, Busy, Waiting} x 2 = {Idle, Busy} x 3 = {Idle, Carrying, Returning} r t t W B I x t r 2 r t B I t d t t x 3 R C I t Page 54

55 Discrete Event Systems Example of DES: Events: new part M 1 R M 2... Idle Idle Idle a 1 - loads part in M 1 d 1 - ends part processing in M 1 r 1 - loads manipulator r 2 - unloads manipulator and loads M 2 d 2 - ends part processing in M 2 r 3 - manipulator at base reject part Busy r 1 Carrying r 2 Busy d 1 d 2 Waiting M 1 M 2 Returning R r 3 Page 55

Industrial Automation de Processos Industriais)

Industrial Automation de Processos Industriais) Industrial Automation (Automação de Processos Industriais) (Sequential Function Chart) http://users.isr.ist.utl.pt/~jag/courses/api1213/api1213.html Slides 2010/2011 Prof. Paulo Jorge Oliveira Rev. 2011-2013